View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server PITT-97-447; CMU-HEP-97-14; DOE-ER/40682-139;LPTHE-97- NON-EQUILIBRIUM EVOLUTION OF A `TSUNAMI': DYNAMICAL SYMMETRY BREAKING (a) (b) (c) Daniel Boyanovsky , Hector J. de Vega , Richard Holman , (c) (d) S. Prem Kumar , Rob ert D. Pisarski (a) Department of Physics and Astronomy, University of Pittsburgh, PA. 15260,U.S.A (b) LPTHE, Universite Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), Tour 16, 1 er. etage, 4, Place Jussieu 75252 Paris, Cedex 05, France (c) Department of Physics, Carnegie Mel lon University, Pittsburgh, P.A. 15213, U.S.A. (d) Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, U.S.A. Abstract We prop ose to study the non-equilibrium features of heavy-ion collisions by following the evolution of an initial state with a large numb er of quanta with ~ a distribution around a momentum jk j corresp onding to a thin spherical shel l 0 2 2 ~ in momentum space, a `tsunami'. An O (N ) ( ) mo del eld theory in the large N limit is used as a framework to study the non-p erturbative asp ects of the non-equilibrium dynamics including a resummation of the e ects of the medium (the initial particle distribution). In a theory where the symmetry is sp ontaneously broken in the absence of the medium, when the initial number of particles p er correlation volume is chosen to b e larger than a critical value the medium e ects can restore the symmetry of the initial state. We show that if one b egins with sucha symmetry-restored, non-thermal, initial state, non-p erturbative e ects automatically induce spino dal instabilities leading to a dynamical breaking of the symmetry. As a result there is explosive particle pro duction and a redistribution of the particles towards low momentum due to the nonlinearity of the dynamics. The asymptotic b ehavior displays the onset of Bose condensation of pions and the equation of state at long times is that of an ultrarelativistic gas although the momentum distribution is non-thermal. 11.10.-z,11.30.Qc,1 1.1 5. Tk Typ eset using REVT X E 1 I. INTRODUCTION The Relativistic Heavy Ion Collider (RHIC) at Bro okhaven and the Large Hadron Collider (LHC) at CERN will provide an unprecedented range of energies and luminosities that will hop efully prob e the Quark-Gluon Plasma and Chiral Phase transitions. The basic picture of the ion-ion collisions in the energy ranges prob ed by these accelerators as seen in the center- of-mass frame (c.m.), is that of two highly Lorentz-contracted `pancakes' colliding and leaving a `hot' region at mid-rapidity with a high multiplicity of secondaries [1]. At RHIC for Au + Au 26 2 central collisions with typical luminosityof10 =cm :s and c.m. energy 200GeV=n n,a multiplicity of 500-1500 particles p er unit rapidity in the central rapidity region is exp ected 27 2 [2{4]. At LHC for head on Pb + Pb collisions with luminosity 10 =cm :s at c.m. energy 5TeV=n n, the multiplicityofcharged secondaries will b e in the range 2000 8000 p er unit rapidity in the central region [4]. At RHIC and LHC typical estimates [1{6] of energy 3 densities and temp eratures near the central rapidity region are " 1 10 GeV=fm ;T 0 300 900 MeV . Since the lattice estimates [4,6] of the transition temp eratures in QCD, b oth for the QGP and Chiral phase transitions are T 160 200 MeV, after the collision the central region c will b e at a temp erature T >T . In the usual dynamical scenario that one [1] envisages, the c initial state co ols o via hydro dynamic expansion through the phase transition down to a freeze-out temp erature, estimated to be T 100 MeV [7], at which the mean free-path of F the hadrons is comparable to the size of the expanding system. The initial state after the collision is strongly out of equilibrium and there are very few quantitative mo dels to study its subsequent evolution. There are p erturbative and non- p erturbative phenomena that contribute to the pro cesses of thermalization and hadroniza- tion. The p erturbative (hard and semihard) asp ects are studied via parton cascade mo dels which assume that at large energies the nuclei can b e resolved into their partonic constituents and the dynamical evolution can therefore be tracked by following the parton distribution functions through the p erturbative parton-parton interactions [8{12]. Parton cascade mo dels (including screening corrections to the QCD parton-parton cross sections) predict that ther- malization o ccurs on time scales 0:5fm=c [13]. After thermalization, and provided that the mean-free path is much shorter than the typical interparticle separation, further evolution of the plasma can be describ ed with b o ost-invariant relativistic hydro dynamics [1,14]. The details of the dynamical evolution between the parton cascade through hadronization, and eventual description via hydro dynamics is far from clear but will require a non-p erturbative treatment. The non-p erturbative asp ects of particle pro duction and hadronization typically envisage a ux-tub e of strong color-electric elds, in which the eld energy leads to pro duc- tion of qq pairs [15,16]. Recently the phenomenon of pair pro duction from strong electric elds in b o ost-invariant co ordinates was studied via non-p erturbative metho ds that address the non-equilibrium asp ects and allow a comparison with hydro dynamics [17]. The dynamics near the phase transition is even less understo o d and involves physics be- yond the realm of p erturbative metho ds. For instance, considerable interest has b een sparked recently by the p ossibility that disoriented chiral condensates (DCC's) could form during the 2 evolution of the QCD plasma through the chiral phase transition [18]- [23]. Ra jagopal and Wilczek [24] have argued that if the chiral phase transition o ccurs strongly out of equilibrium, spino dal instabilities [25] could lead to the formation and relaxation of large pion domains. This phenomenon could provide a striking exp erimental signature of the chiral phase transi- tion and could provide an explanation for the Centauro and anti-Centauro (JACEE) cosmic ray events [26]. An exp erimental program is underway at Fermilab to search for candidate events [27,28]. Most of the theoretical studies of the dynamics of the chiral phase transition and the p ossibility of formation of DCC's have fo cused on initial states that are in lo cal thermo dynamic equilibrium (LTE) [29]- [32]. We prop ose to study the non-equilibrium asp ects of the dynamical evolution of highly excited initial states by relaxing the assumption of initial LTE (as would be appropriate for the initial conditions in a heavy-ion collision). Consider, for example, a situation where the relevant quantum eld theory is prepared in an initial state with a particle distribution ~ ~ ~ sharply p eaked in momentum space around k and k where k is a particular momentum. 0 0 0 This con guration would b e envisaged to describ e two `pancakes' or `walls' of quanta moving ~ in opp osite directions with momentum jk j. In the target frame this eld con guration 0 would b e seen as a `wall' of quanta moving towards the target and hence the name `tsunami' [33]. Such an initial state is out of equilibrium and under time evolution with the prop er interacting Hamiltonian, non-linear e ects should result in a redistribution of particles, as well as particle pro duction and relaxation. The evolution of this strongly out of equilibrium initial state would b e relevant for understanding phenomena such as formation and relaxation of chiral condensates. Starting from such a state and following the complete evolution of the system thereon, is clearly a formidable problem even within the framework of an e ective eld theory such as the linear -mo del. In this article we consider an even more simplistic initial condition, where the o ccupation numb er of particles is sharply lo calized in a thin spherical shel l in momentum space around ~ a momemtum jk j, i.e. a spherical ly symmetric version of the `tsunami' con guration. The 0 reason for the simpli cation is purely technical since spherical symmetry can be used to reduce the number of equations. Although this is a simpli cation of the idealized problem, it will be seen b elow that the features of the dynamics contain the essential ingredients to help us gain some understanding of more realistic situations. 4 2 We consider a weakly coupled theory ( 10 ) with the elds in the vector representation of the O (N ) group. Anticipating non-p erturbative physics, we study the dynamics consistently in the leading order in the 1=N expansion which will allow an analytic treatment as well as anumerical analysis of the dynamics. The pion wall scenario describ ed ab ove is realized by considering an initial state describ ed ~ by a Gaussian wave functional with a large numb er of particles at jk j and a high densityis 0 achieved by taking the number of particles per correlation volume to be very large. As in nite temp erature eld theory, a resummation along the lines of the Braaten and Pisarski [34] program must b e implemented to takeinto account the non-p erturbative asp ects of the physics in the dense medium.